Oblique Triangle Solver

Oblique Triangle Solver – Solve Any Non-Right Triangle

An oblique triangle is any triangle that does not contain a 90° angle. Because the Pythagorean theorem does not apply, solving oblique triangles requires more general tools: the Law of Sines and the Law of Cosines. This solver handles all five standard input combinations — SSS, SAS, ASA, AAS, and SSA — and computes every property of the triangle including all sides, all angles, area, perimeter, inradius, and circumradius.

The Five Triangle Cases

Any triangle can be uniquely determined (or shown to be impossible) from exactly three known measurements, provided at least one is a side length. The five cases are named after the combination of sides (S) and angles (A) given:

• SSS — all three sides known. The Law of Cosines finds each angle: cos(A) = (b² + c² − a²) / (2bc).
• SAS — two sides and the angle between them. The Law of Cosines finds the missing side: c² = a² + b² − 2ab·cos(C).
• ASA — two angles and the side between them. The third angle follows from A + B + C = 180°; the Law of Sines finds the other two sides.
• AAS — two angles and a non-included side. Similar to ASA; the third angle is computed first, then the Law of Sines gives the remaining sides.
• SSA — two sides and a non-included angle. This is the ambiguous case: zero, one, or two triangles may be valid. The solver detects all possibilities and displays both solutions when two exist.

When to Use Law of Sines vs Law of Cosines

The Law of Cosines is the primary tool for SSS and SAS cases. It directly relates three sides to one angle without requiring another angle to be known:

a² = b² + c² − 2bc·cos(A)

The Law of Sines is used for ASA, AAS, and SSA cases, where at least one angle and its opposite side are known:

a / sin(A) = b / sin(B) = c / sin(C)

Using the Law of Cosines for SSS and SAS (instead of the Law of Sines) avoids potential numerical instability near 0° and 180° and sidesteps the SSA ambiguity in those cases.

The Ambiguous SSA Case

The SSA configuration is unique because the given information does not always pin down a unique triangle. Given angle A, side a (opposite A), and side b, the altitude from vertex A has length h = b·sin(A). The number of valid triangles depends on how a compares to h and b:

• If a < h: side a is too short to reach the opposite base — no triangle exists.
• If a = h: side a is exactly the altitude — one right triangle exists.
• If h < a < b: two distinct triangles satisfy the data — the solver shows both with their respective properties and diagrams.
• If a ≥ b (or angle A ≥ 90°): only one triangle exists.

Derived Triangle Properties

Once all three sides a, b, c are known, additional properties follow directly:

• Area — via Heron's formula: Area = √(s·(s−a)·(s−b)·(s−c)), where s = (a+b+c)/2 is the semi-perimeter.
• Inradius (inscribed circle) — r = Area / s. The inradius is tangent to all three sides.
• Circumradius (circumscribed circle) — R = (a·b·c) / (4·Area). Every triangle has exactly one circumscribed circle passing through all three vertices.
• Triangle type— classified as Acute (all angles < 90°), Obtuse (one angle > 90°), Right (one angle = 90°), or Equilateral (all angles = 60°).

Practical Applications

Oblique triangle solving appears in many real-world fields. Surveyors use AAS and ASA triangulation to measure land distances without direct access to every point. Engineers and architects apply the Law of Cosines in SAS configurations when designing trusses, roof pitches, and structural layouts. Navigation relies on oblique triangle calculations for bearing and distance problems between waypoints. Students studying trigonometry encounter all five cases in coursework and standardized exams.

Using This Calculator

Select the case that matches your known values using the five buttons at the top. Enter your measurements in the input fields — side lengths in any unit, angles in degrees or radians. The results panel updates instantly with all computed sides, angles, and derived properties. A labeled SVG diagram visualizes the solved triangle. For the SSA case, two diagrams appear side by side when two solutions exist. Use the copy button to export any result set as plain text.